The Newman algorithm for constructing polynomials with restricted coefficients and many real roots

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets Eₖ of admissible coefficients, we construct a polynomial Pₙ(x)=1+∑ₖ₌₁ⁿεₖ xᵏ, εₖ∈ Eₖ, with at least c√n distinct roots in $[0,1]$, which matches the classical upper bound up to the value of the constant $c>0$. Our sufficient conditions cover the Littlewood (Eₖ=\-1,1\) and Newman (Eₖ=\0,(-1)ᵏ\) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence Eₖ is periodic.